CN112270092A - Multi-iteration-point artificial wave response spectrum fitting method for self-recognition of hard points - Google Patents

Multi-iteration-point artificial wave response spectrum fitting method for self-recognition of hard points Download PDF

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CN112270092A
CN112270092A CN202011171628.XA CN202011171628A CN112270092A CN 112270092 A CN112270092 A CN 112270092A CN 202011171628 A CN202011171628 A CN 202011171628A CN 112270092 A CN112270092 A CN 112270092A
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谢皓宇
仉文岗
黎泳钦
顾鑫
邹育麟
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Abstract

In order to avoid the increase of errors caused by stubborn points in the process of fitting the artificial seismic waves, the invention provides a multi-iteration point artificial wave reaction spectrum fitting method for self identification of the stubborn points, which comprises the steps of firstly converting a designed reaction spectrum into a power spectral density function by using an empirical formula, secondly combining a random phase spectrum, generating a zero-mean stable Gaussian process by using fast Fourier inverse transformation, and multiplying the zero-mean stable Gaussian process by an envelope function to obtain artificial waves; and then, a Fourier amplitude spectrum is iterated through the obtained artificial wave reaction spectrum value, the frequency sampling points are corrected, the corrected frequency sampling points are iterated to obtain a Fourier amplitude spectrum corresponding to the artificial wave, when the error between the artificial wave reaction spectrum frequency sampling points and the frequency sampling points of the design spectrum is small enough, convergence is judged, and the fitting of the seismic wave is completed. The artificial seismic waves generated by the method reduce errors caused by stubborn points and improve the convergence precision of the artificial seismic waves; thereby reducing the error of the artificially synthesized seismic waves.

Description

Multi-iteration-point artificial wave response spectrum fitting method for self-recognition of hard points
Technical Field
The invention relates to the technical field of artificial seismic wave synthesis, in particular to a multi-iteration-point artificial wave reaction spectrum fitting method for self-identification of stubborn points.
Background
In the performance-based earthquake-proof design concept, the fields of building structure earthquake-proof design, bridge structure earthquake-proof design, nuclear power plant and equipment earthquake-proof design and the like are included, time-course dynamic analysis is an indispensable earthquake-proof performance evaluation means, and acquisition of input earthquake waves in the time-course dynamic analysis process has two common ways, one is that a strong seismograph is directly used for recording strong earthquake in the nature, and the other is that earthquake waves are artificially synthesized through calculation. At present, actual earthquake records are limited, particularly strong earthquake records are rare, and fitting of artificial earthquake waves according to a design response spectrum is a very important problem when power analysis is carried out in the process of carrying out structural earthquake-resistant design or earthquake-resistant performance evaluation by technical personnel in the industries such as engineers, scholars and the like.
The synthesis of artificial seismic waves generally adopts a frequency domain method, also called a trigonometric function superposition method, which is the most widely used method for designing response spectrum fitting artificial seismic waves (Hu 32895; sages, who is trained; artificial seismic response spectrum fitting considering phase spectrum [ J ]. seismic engineering and engineering vibration, 1986(2): 39-53.). The frequency domain method superimposes Fourier components with different amplitudes and phases into artificial waves by utilizing inverse Fourier transform, then calculates corresponding reaction spectrums, iterates according to errors between the reaction spectrums and the designed reaction spectrums, and judges the artificial waves to be converged until the errors are small enough, for example, Chinese patent CN109002678B discloses an artificial earthquake simulation method based on hydropower engineering hydraulic structure earthquake-resistant design. In practical application, the method has frequency control points which cannot improve convergence accuracy due to the fact that the iteration times are increased, and the points are called stubborn points; due to the existence of these stubborn points, convergence occurs in the fitting process, and further, the error of the artificially synthesized seismic wave is large.
Disclosure of Invention
The invention provides a multi-iteration point artificial wave response spectrum fitting method for self-identification of stubborn points, which can automatically identify stubborn points in the fitting process and avoid the increase of errors of the stubborn points in the fitting process aiming at the stubborn points in the process of fitting the artificial seismic waves by a frequency domain method.
In order to achieve the aim, the method for fitting the multi-iteration point artificial wave reaction spectrum by self-recognition of the stubborn points comprises the following steps:
s100: using empirical formula to design reaction spectrum SDConversion to power spectral density function
Figure BDA0002747467310000012
The empirical formula is:
Figure BDA0002747467310000011
Ω＝[ω1,ω2,ω3,……,ωK-1,ωK] (2)
in the formula (1), zeta is damping ratio, gamma is transcendental probability, T is seismic wave duration,
in the formula (2), omega is the frequency sampling point set of the target reaction spectrum, K is the number of sampling points, and omegaKThe frequency corresponding to the K point;
s200: calculating a Fourier amplitude spectrum of the artificial seismic waves according to the power spectral density function:
Figure BDA0002747467310000021
in the formula (3), A (ω)k) The Fourier amplitude spectrum corresponding to the artificial wave is obtained, and delta omega is the sampling interval of the frequency spectrum;
s300: combining the Fourier amplitude spectrum obtained by calculation in the step S200 with a random phase spectrum, generating a zero-mean stable Gaussian process by using inverse fast Fourier transform, and multiplying the zero-mean stable Gaussian process by an envelope function to generate an artificial wave:
Figure BDA0002747467310000022
Figure BDA0002747467310000023
in the formula (4), FFT-1Is the inverse operator of the fourier transform,
Figure BDA0002747467310000024
in the form of a random phase spectrum,
Figure BDA0002747467310000025
a stable Gaussian process is adopted;
in formula (5), I (t) is a time-course envelope function,
Figure BDA0002747467310000026
is artificial wave;
s400: the artificial wave obtained in step S300 is subjected to a reaction spectrum value, a fourier amplitude spectrum is iterated by squaring the ratio between the designed reaction spectrum and the calculated reaction spectrum,
Figure BDA0002747467310000027
in the formula (6), An(ωk) Obtaining a Fourier amplitude spectrum corresponding to the artificial wave for the nth iteration, Sn(ωk) Response spectra corresponding to the nth iteration artificial wave, SD(ωk) Designing a reaction spectrum;
s500: correcting the frequency sampling point, wherein the correction condition is an expression (7);
Figure BDA0002747467310000028
s600: iterating the corrected frequency sampling points to obtain a Fourier amplitude spectrum corresponding to the artificial wave,
Figure BDA0002747467310000029
in the formula (8), ωjC is an iteration influence constant for the frequency domain sampling point needing iteration correction.
S700: when the error between the reaction spectrum frequency sampling point of the artificial wave and the frequency sampling point of the design spectrum is small enough, the convergence is judged, and the fitting of the seismic wave is completed:
Figure BDA0002747467310000031
in the formula, SN(ωk) After N iterations, the response spectrum of the artificial seismic waves in convergence is shown, and sigma is a convergence judgment constant.
The artificial seismic waves generated by the method can automatically identify stubborn points in the process of fitting the artificial seismic waves corresponding to the response spectrum by a frequency domain method, modify the traditional single-point iteration into a multi-point iteration enveloped by a normally distributed probability density function, and solve the problem that the stubborn points are dispersed or can not be converged in the existing calculation method by utilizing the iteration influence constant. The error caused by stubborn points is reduced, and the convergence precision of the artificial seismic waves is improved; thereby reducing the error of the artificially synthesized seismic waves.
Drawings
FIG. 1 is a schematic diagram of frequency domain stubborn points occurring during an iteration process;
FIG. 2 is a schematic diagram comparing the present invention with a conventional iterative method;
FIG. 3 is a design reaction spectrum selected in the section "seismic design Specification for road and bridge" (JTG/T2231-01-2020);
FIG. 4 is a comparison graph of the reaction spectra obtained by the method of the present application and the prior art after the first set of Fourier phase spectrum iterations, respectively, in the example, where the left side is the reaction spectrum after the iteration performed by the method of the present application, and the right side is the reaction spectrum after the iteration performed by the prior art;
FIG. 5 is a comparison graph of the reaction spectra obtained by the method of the present application and the prior art after a second set of Fourier phase spectrum iterations, respectively, in the example, where the left side is the reaction spectrum after the iteration performed by the method of the present application, and the right side is the reaction spectrum after the iteration performed by the prior art;
FIG. 6 is a comparison graph of reaction spectra obtained by the method of the present application and the prior art after a third set of Fourier phase spectrum iterations, respectively, in the example, where the left side is the reaction spectrum after the iteration performed by the method of the present application, and the right side is the reaction spectrum after the iteration performed by the prior art;
Detailed Description
While the following description details certain exemplary embodiments which embody features and advantages of the invention, it will be understood that various changes may be made in the embodiments without departing from the scope of the invention, and that the description and drawings are to be regarded as illustrative in nature and not as restrictive.
Referring to fig. 1 and fig. 2, the method for fitting the multi-iteration point artificial wave reaction spectrum for self-identification of hard points in the application comprises the following steps,
s100: using empirical formula to design reaction spectrum SDConversion to power spectral density function
Figure BDA0002747467310000032
The empirical formula is:
Figure BDA0002747467310000041
Ω＝[ω1,ω2,ω3,……,ωK-1,ωK] (2)
in the formula (1), zeta is damping ratio, gamma is transcendental probability, T is duration of seismic wave,
in the formula (2), omega is a frequency sampling point set of the target reaction spectrum, and K is the number of sampling points;
s200: calculating a Fourier amplitude spectrum of the artificial seismic waves according to the power spectral density function:
Figure BDA0002747467310000042
in the formula (3), A (ω)k) The Fourier amplitude spectrum corresponding to the artificial wave is obtained, and delta omega is the sampling interval of the frequency spectrum; further, the sampling interval Δ ω is constantAnd (4) counting. Because the invention relates to the use of Fast Fourier Transform (FFT), the sampling points of the frequency spectrum need to be complemented according to the requirement of the FFT, and the value of delta omega and the number of the sampling points 2 which are considered to be defined and are used for fitting the artificial seismic waves at the momentNAnd the sampling interval Δ t determines:
Figure BDA0002747467310000043
s300: combining the Fourier amplitude spectrum obtained by calculation in the step S200 with a random phase spectrum, generating a zero-mean stable Gaussian process by using inverse fast Fourier transform, and multiplying the zero-mean stable Gaussian process by an envelope function to generate an artificial wave:
Figure BDA0002747467310000044
Figure BDA0002747467310000045
in the formula (4), FFT-1Is the inverse operator of the fourier transform,
Figure BDA0002747467310000046
in the form of a random phase spectrum,
Figure BDA0002747467310000047
a stable Gaussian process is adopted;
in formula (5), I (t) is a time-course envelope function,
Figure BDA0002747467310000048
is artificial wave;
s400: calculating a reaction spectrum value of the artificial wave obtained in the step S300, and iterating a fourier amplitude spectrum by a square of a ratio between the designed reaction spectrum and the calculated reaction spectrum:
Figure BDA0002747467310000049
in the formula (6), An(ωk) Obtaining a Fourier amplitude spectrum corresponding to the artificial wave for the nth iteration, Sn(ωk) Response spectra corresponding to the nth iteration artificial wave, SD(ωk) Designing a reaction spectrum;
s500: correcting the frequency sampling point, and further correcting on the basis of the formula 6 in order to automatically identify and avoid stubborn points:
Figure BDA0002747467310000051
formula 7 gives the iterative correction condition, that is, the iterative process starts to diverge, which is also a characteristic of the stubborn point in the artificial wave frequency domain method fitting process, and the automatic identification of the stubborn point can be realized through formula 7.
S600: iterating the corrected frequency sampling points to obtain a Fourier amplitude spectrum corresponding to the artificial wave,
Figure BDA0002747467310000052
in the formula (8), ωjC is an iteration influence constant for the frequency domain sampling point needing iteration correction.
S700: when the error between the reaction spectrum frequency sampling point of the artificial wave and the frequency sampling point of the design spectrum is small enough, the convergence is judged, and the fitting of the seismic wave is completed:
Figure BDA0002747467310000053
in the formula, SN(ωk) After N iterations, the response spectrum of the artificial seismic waves is converged, sigma is a convergence judgment constant,
further, the convergence determination constant σ is 0.01.
Further, the iterative influence constant takes the following values:
Figure BDA0002747467310000054
formula 9 defines the value of c at ωkReaction spectrum acceleration a and omega corresponding to frequencykThe response acceleration a of the vibrator at the moment of maximum acceleration under excitationωkThe same direction is taken as 2, and the opposite direction is taken as-2.
Further, the value range of the frequency sampling points is as follows:
j＝(k-3),(k-2),(k-1),k,(k+1),(k+2),(k+3) (10)
formula 10 gives ωjI.e. 7 sampling frequency points of the stubborn point itself and the surrounding sampling points. According to the method, the reaction spectrum at the frequency of the stubborn point is finally influenced by adjusting the Fourier amplitude spectrum of the frequency control point near the stubborn point frequency, and the influence has a marginal effect, namely the farther the control frequency is away from the stubborn point, the smaller the influence is, so that 3 points before and after the stubborn point control frequency are selected for the frequency sampling point in the method; the accuracy of the calculation result can be ensured, and the calculation amount can be reduced.
In the examples of this application, the design response spectra specified in the section "design Specification for earthquake resistance of road bridges" (JTG/T2231-01-2020) was used, with the parameters selected in the following table.
TABLE 1
Figure BDA0002747467310000061
According to the standard requirements, the corresponding reaction spectrum values of the frequency control points are shown in table 2, and the reaction spectra are shown in figure 3.
TABLE 2
Figure BDA0002747467310000062
Figure BDA0002747467310000071
In the specific implementation, the same 3 sets of fourier phase spectra are used, the method of the present invention and the frequency domain fitting method in the prior art are used for 3 iterations respectively, and 3 artificial seismic waves are generated, wherein the fitting effect refers to fig. 4 to fig. 6 in detail. It can be seen that, in 3 groups of different Fourier phase spectrums, the fitting precision difference of different Fourier phase spectrums is larger, but in the same group of phase spectrums, the fitting precision of the method is obviously better than that of a frequency domain fitting method in the prior art, and no matter in an ascending section, a platform section or a descending section of a reaction spectrum, the stubborn point of the method is self-identified and avoided and the error is reduced.

Claims (5)

1. The method for fitting the multi-iteration-point artificial wave reaction spectrum for self-recognition of the stubborn points is characterized by comprising the following steps:
s100: using empirical formula to design reaction spectrum SDConversion to power spectral density function
Figure FDA0002747467300000011
The empirical formula is:
Figure FDA0002747467300000012
Ω＝[ω1，ω2，ω3，……，ωK-1，ωK] (2)
in the formula (1), zeta is damping ratio, gamma is transcendental probability, T is seismic wave duration,
in the formula (2), omega is the frequency sampling point set of the target reaction spectrum, K is the number of sampling points, and omegaKThe frequency corresponding to the K point;
s200: calculating a Fourier amplitude spectrum of the artificial seismic waves according to the power spectral density function:
Figure FDA0002747467300000013
in the formula (3), the reaction mixture is,A(ωk) The Fourier amplitude spectrum corresponding to the artificial wave is obtained, and delta omega is the sampling interval of the frequency spectrum;
s300: combining the Fourier amplitude spectrum obtained by calculation in the step S200 with a random phase spectrum, generating a zero-mean stable Gaussian process by using inverse fast Fourier transform, and multiplying the zero-mean stable Gaussian process by an envelope function to generate an artificial wave:
Figure FDA0002747467300000014
Figure FDA0002747467300000015
in the formula (4), FFT-1Is the inverse operator of the fourier transform,
Figure FDA0002747467300000016
in the form of a random phase spectrum,
Figure FDA0002747467300000017
a stable Gaussian process is adopted;
in formula (5), I (t) is a time-course envelope function,
Figure FDA0002747467300000018
is artificial wave;
s400: the artificial wave obtained in step S300 is subjected to a reaction spectrum value, a fourier amplitude spectrum is iterated by squaring the ratio between the designed reaction spectrum and the calculated reaction spectrum,
Figure FDA0002747467300000019
in the formula (6), An(ωk) Obtaining a Fourier amplitude spectrum corresponding to the artificial wave for the nth iteration, Sn(ωk) Response spectra corresponding to the nth iteration artificial wave, SD(ωk) Designing a reaction spectrum;
s500: correcting the frequency sampling point, wherein the correction condition is an expression (7);
Figure FDA00027474673000000110
s600: iterating the corrected frequency sampling points to obtain a Fourier amplitude spectrum corresponding to the artificial wave,
Figure FDA0002747467300000021
in the formula (8), ωjC is an iteration influence constant for a frequency domain sampling point needing iteration correction;
s700: when the error between the reaction spectrum frequency sampling point of the artificial wave and the frequency sampling point of the design spectrum is small enough, the convergence is judged, and the fitting of the seismic wave is completed:
Figure FDA0002747467300000022
in the formula, SN(ωk) After N iterations, the response spectrum of the artificial seismic waves in convergence is shown, and sigma is a convergence judgment constant.
2. The method for fitting a multi-iteration point artificial wave reaction spectrum for stubborn point self-identification according to claim 1, wherein the sampling interval Δ ω in the step S200 is a constant.
3. The method for fitting the obstinate point self-identified multi-iteration-point artificial wave response spectrum according to claim 1, wherein the iteration influence constant takes the value as follows:
Figure FDA0002747467300000023
formula (9)) In which a is omegakResponse spectrum acceleration corresponding to the frequency; a isωkIs omegakThe response acceleration of the vibrator at the moment of maximum acceleration under the excitation of the Fourier component of (1).
4. The method for fitting the obstinate point self-identified multi-iteration-point artificial wave response spectrum according to claim 1, wherein the value range of the frequency sampling points is as follows: j ═ k-3, (k-2), (k-1), k, (k +1), (k +2), and (k + 3).
5. The method of claim 1, wherein the convergence criterion constant σ is 0.01.
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